The Red Sox had designated Carp for assignment with the goal of trading him.
To make room for Carp on the 40-man roster, the Rangers moved right-handed starter Alexi Ogando to the 60-day disabled list.
Prior to the trade deadline, Carp had asked the Red Sox for a trade because he wanted more playing time.
"For his desire to get more additional at-bats, and I can't say the role he's going into there, but we wish him well," Red Sox manager John Farrell said Sunday.
Carp played a total of 128 games in parts of two seasons for Boston. In 2013, he hit .296 with nine home runs and 43 RBIs in a career-high 86 games for the Red Sox. He appeared in six games and made one start in the postseason, going 0-for-8 with one RBI.
This season, he played 42 games and had a .198 average and nine RBIs.
Information from ESPNBoston.com's Joe McDonald contributed to this report.